Assumption REF  is a basic identifiability condition in a functional linear model and these conditions are discussed in detail in [1]} and [2]}. Assumption REF  corresponds to Assumption A of [2]} which are basic conditions that ensure the statistical problem is correctly posed. For our purposes, however, we replace [4]}'s finite fourth moment assumption on the \(\xi _l\) 's with a stronger finite sixth moment assumption. Assumption REF  corresponds to Assumption F of [2]} which effectively says the sequence of functions \(\lbrace f_n\rbrace \)  should behave like \(f_n(x) \approx 1 / x\)  when \(n\)  is sufficiently large. Assumption REF  is new: it says \(\lbrace  a_n \rbrace \)  is a regularization that tends to zero, and more importantly, tends to zero faster than the reciprocal of the eigenvalues tending to infinity. Assumption REF  will be used to ensure the empirical eigenvectors of the empirical covariance operator uniformly converges in probability to the population eigenvectors of population covariance operator.
